Extinction of intra prey induced by catastrophic shift in a Lesile–Gower intraguild predation model

Abstract

In this letter, we revisit a Lesile–Gower intraguild predation model proposed by Safuan et al. in the paper Safuan et al. (2013). It was shown there as the biotic resource enrichment parameter $\gamma$ varies, the model can undergo a transcritical bifurcation which might explain two alternative scenarios: one is the coexistence of three populations, and the other is the extinction of the intra prey. That is, the survival of intra prey population is solely determined by the biotic resource enrichment. In fact, using the same parameter $\gamma$ as bifurcation parameter, the model can also undergo a saddle–node bifurcation at some critical value ${\gamma }_{SN}$, which might explain another two alternative scenarios: one is the bistability between a positive equilibrium and an intra prey extinction one, and the other is the extinction of the intra prey. This means that the intra prey species may undergo a catastrophic shift when $\gamma$ increases passing through ${\gamma }_{SN}$. This is established by proving the existence of positive equilibria and determining their stability, theoretically and numerically.